LECTURE 11 A MODEL OF LIQUIDITY PREFERENCE

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1 LECTURE 11 A MODEL OF LIQUIDITY PREFERENCE JACOB T. SCHWARTZ EDITED BY KENNETH R. DRIESSEL Abstract. We construct a model of production and exchange within which a decision to expand or to reduce inventories can be studied as an optimization problem. We find that optimum adjustment to the conditions of the constructed model may lead to decisions to reduce inventory in the manner assumed in the preceding lectures. 1. Establishing the Model A key element in the cycle-theory presented in Lectures 5 10 was the decision to reduce inventories which we built into our models, basing ourselves on reasonable empirical grounds, but which we left largely unanalyzed. When manufacturers reduce inventories they free resources for other uses: how does this affect the conclusions developed in the preceding lectures? In the present lecture and in the one which follows, we shall attempt to illuminate this question. Our method will be to construct a model of production and exchange within which a decision to expand or to reduce inventories can be studied as a problem of the optimization type familiar from efficiency economics. We will find that optimum adjustment to the conditions of the model to be constructed may lead to decisions to reduce inventory in the manner assumed in the preceding lectures. In this way, we will confirm our earlier conclusions and establish them upon a more ample basis. Our model is as follows: we take, as usual, an economy involving n material commodities C 1,, C n, and L + 1 labor commodities 2010 Mathematics Subject Classification. Primary 91B55; Secondary 91B08. Key words and phrases. Liquidity Preference, Business Cycle Theory. 1

2 2 JACOB T. SCHWARTZ C 0,, C L. Again, as usual, we take the input-output matrix to be π i,j, where 1 i n and L j n. In addition, we introduce a money commodity C n+1 which cannot be produced; this money commodity is taken to consist of a collection of identical, infinitely divisible, legal-tender bonds issued by a central authority and bearing interest, at a rate 100s per cent per day or planning period, payable in additional bonds. Thus, the possessor of x bonds will at the end of one day s uninterrupted possession be awarded sx additional bonds. We shall assume that s 0. Let the (instantaneous) price of one unit of commodity C k be p k ; we assume that all prices are positive. Revising our otherwise arbitrary physical units of each commodity, we may suppose that p j = 1 for all j; these price units will be used throughout the present lecture unless the contrary is explicitly indicated. With this normalization, the essential features of the present model may be described as follows. Each individual entrepreneur is assumed to be in possession, at the start of a typical day, of holdings of the commodities C 1,, C n+1 in certain amounts b 1,, b n+1. This being his situation, the entrepreneur is assumed to have the option of dividing his holding of bonds into n + 1 parts, using the jth part, 1 j n, for the purchase of the inputs to the production of C j in appropriate proportions and then immediately using these inputs completely, thereby producing a certain well determined number of units of C j. The (n + 1)st part into which the entrepreneur s holding of bonds is divided is assumed to be retained in order to earn interest at the given rate s. Moreover, the entrepreneur is assumed to have the additional option of dividing his holding of each sort of commodity into two parts, one of which is retained in storage, the other of which is offered for sale. We assume that conditions in the C j -market are such that of each unit of C j offered for sale, σ j units of C j will be sold, and 1 σ j units will remain unsold; thus σ j is a parameter between 0 and 1 which measures the instantaneous state of the market for the commodity C j, as reflected in the rate of stock turnover which an optional sales policy is capable of achieving. In order to take account within our model of style

3 11. A MODEL OF LIQUIDITY PREFERENCE 3 obsolescence and technical obsolescence, spoilage, and fixed overhead costs, we assume that each unit of C j which is either retained or unsold decays after one day to a certain number τ j of units; here 0 τ j < 1. The coefficient τ j then is a general spoilage and carrying charge loss parameter. Let the number of units of commodity C j which can be produced by expending one bond on the necessary inputs in correct proportion and using them to produce C j be η j ; η j is of course determined not merely by the input-output matrix but also by the prices; we have 0 < η j. Then, starting with holdings of C 1,, C n+1 equal to the components of the vector b = [b 1,, b n+1 ], and dividing it in accordance with the above-described options into a sum of nonnegative terms (11.1) b = [b 11 + b 12, b 21 + b 22,, b n1 + b n2, b n+1,1 + + b n+1,n+1 ], the entrepreneur will upon exercise of these options find himself at the end of the day in possession of inventories of C 1,, C n+1 equal to the components of the following vector c: (11.2) c = [τ 1 b 11 + τ 1 (1 σ 1 )b 12 + η 1 b n+1,1, τ 2 b 21 + τ 2 (1 σ 2 )b 22 + η 2 b n+1,2,, τ n b n1 + τ n (1 σ n )b n2 + η n b n+1,n, σ 1 b 12 + σ 2 b σ n b n2 + (1 + s)b n+1,n+1 ]. Let φ(b) denote the set of all vectors c which can be derived from a given vector b after the fashion of (11.1) (11.2); then φ is a (one-tomany, i.e., many-valued) transformation defined in the positive orthant. Of course, φ depends upon the constants σ j reflecting the instantaneous state of the market for C j and upon the instantaneous prices p j ; were we at the present moment aiming at the direct integration of the transformation φ into a dynamic model of prices and production, we would have to deal at once with this dependence. Our idea, however, is somewhat different. We wish to analyze an entrepreneur s projection of his future operations, assuming, as in the cycle-theory of Lectures 5, 6, and 7, that these projections are based upon the simplest of

4 4 JACOB T. SCHWARTZ all hypotheses, the hypothesis that market conditions will remain unchanged. In a more complete model the specific action taken by an entrepreneur on any particular day would still be determined by this projection, but the projection would be revised each day as the actual motion of the economy changes the data upon which the entrepreneur bases his projections. We may consider, however, that the phenomena of cyclic expansion and contraction of production which would occur in such a dynamic model have been portrayed with at least qualitative fidelity in Lectures 7 and 10. Thus, in the present lecture, we forego the dynamic problem in order to develop a detailed still photograph of instantaneous market conditions and of the instantaneous optimal adjustment to the conditions which may correspond either to a decision to increase or to reduce inventories. The parameters σ, τ, s of (11.2) should then be taken as descriptive of a given state of business which entrepreneurs expect to persist for a period of time sufficiently long so as to compel them to make at least a temporary adjustment. What we wish to find is this adjustment. For the present, then, it is appropriate for us to take the constants σ j and η j as fixed, and hence to study the iterates of the fixed multivalued transformation φ. We will show in what follows that this study leads in a natural manner to a systematic ranking of inventory positions as more or less favorable. It is then heuristically plausible to suppose that a manufacturer always strives to attain the most favorable position in the sense of this ranking; this assumption enables us to discover his optimal action in the given situation. The analysis leads by a technically complicated path to very simple conclusions; the reader may prefer to read the main statements and preliminary definitions which are to follow immediately, and to ignore the technical mathematics of their proof. 2. Basic Lemmas and Definitions, Statement of Main Results Let b = [b 1,, b n+1 ] be a vector (necessarily of dimension n + 1) describing the inventory holdings of an entrepreneur. It is plain that

5 11. A MODEL OF LIQUIDITY PREFERENCE 5 if b 1 b, the holdings b 1 are definitely preferable to the holdings b. This ordering of vectors is, however, only a partial ordering, in the sense that there exist noncomparable pairs of vectors, i.e., pairs of vectors b 1, b 2 for which both assertions b 1 b 2 and b 2 b 1 are false. We will show in what follows how a linear ordering may be derived from the ordering and the mapping φ, i.e., we will show how to derive a transitive relationship b 1 b 2 such that one of the two alternatives b 1 b 2 or b 2 b 1 necessarily holds for each pair of vectors b 1, b 2. In order to be able to describe this construction, we must first give some preliminary definitions and lemmas. We begin by extending the ordering of vectors to sets of vectors. Definition Let E and F be two sets of vectors. Then we will write E F if for each b 1 F, there exists a b 2 E such that b 2 b 1. The statement E F has then the following heuristic significance: the set of options E is preferable to the set of options F. Next we give a more convenient form to the definition (11.1) (11.2) of the mapping φ. It is evident, in the first place, that the set of vectors c in (11.2) is identical with the set of vectors d such that (11.3) d = [τ 1 (1 σ 1 )b 11 + τ 1 (b 12 + b b 1,n+1 ) + η 1 b n+1,1, τ 2 (1 σ 2 )b 22 + τ 2 (b 21 + b b 2,n+1 ) + η 2 b n+1,2,, τ n (1 σ n )b nn + τ n (b n1 + + b n,n 1 + b n,n+1 ) + η n b n+1,n, σ 1 b 11 + σ 2 b σ n b nn + (1 + s)b n+1,n+1 ] where (11.4) b = [b 11 + b b 1,n+1,b 21 + b b 2,n+1,, b n+1,1 + b n+1,2 + + b n+1,n+1 ],

6 6 JACOB T. SCHWARTZ each of the terms b ij being nonnegative. Thus, if we define (n + 1) (n + 1) matrices M 1,, M n and M n+1 by the equations (11.5) (M k ) i,j = τ j δ ij if 1 i, j n and i, j k; (M k ) i,j = 0 in every other case 1 i, j n + 1 unless (i, j) = (k, n + 1), (n + 1, k), or (k, k); (M k ) k,n+1 = η k ; (M k ) k,k = τ k (1 σ k ); (M k ) n+1,k = σ k ; and (11.6) (M n+1 ) i,j = τ j δ ij, j = 1,, n, i = 1,, n + 1; (M n+1 ) i,n+1 = (1 + s)δ i,n+1, i = 1,, n + 1; we may describe the set φ(b) as follows: (11.7) φ(b) = {d d = M 1 r M n+1 r n+1, r r n+1 = b, r j 0, j = 1,, n + 1}. We will also make use of certain standard set-theoretical notions and notations. If E and F are sets of vectors, we write E F or F E if E is included in F ; we write a E if a is a member of the set E, write φ(e) for the set (11.8) φ(e) = {φ(b) b E}, and write E + F and te for the sets (11.9) E + F = {b 1 + b 2 b 1 E, b 2 F } and (11.10) te = {tb b E}, respectively. The union of E and F will be denoted by E F ; more generally, if {E α } is an arbitrary collection of sets, indexed by an index α, then the union of all the sets E α, i.e., the collection of all elements which lie in at least one of the sets E α, will be written α E α. The following rather trivial lemmas will be quite useful. Lemma Let E F. Then φ(e) φ(f ).

7 11. A MODEL OF LIQUIDITY PREFERENCE 7 Proof: Let c φ(f ), so that there exist r 1,, r n+1 0 such that (11.11) r r n+1 F, M 1 r M n+1 r n+1 = c. Then, since E F, there exists, by Definition 1, a vector b E such that b r r n+1. Put q n+1 = b r 1 r n. Then plainly q n+1 0, r r n + q n+1 = b E, and d = M 1 r M n r n + M n+1 q n+1 c. Since d φ(e), our lemma is proved. Q.E.D. Lemma Let t 0 be a number. Then φ(te) = tφ(e). Proof: Obvious. Q.E.D. We may now make our fundamental definition. Definition Let b and c be two nonnegative vectors. Then we write b c (or c b) if for each ɛ > 0 there exists an n = n(ɛ) such that (1 + ɛ)φ n (b) φ n (c). If both b c and c b are true, we write b c. If b c but b c is false, we write b c. Note that, by Lemmas 2 and 3, the inequality (1 + ɛ)φ n (b) φ n (c) implies that (1 + ɛ)φ m (b) φ m (c) for all m n. In particular, b c implies b c. The relationship b c has the following heuristic significance. The set φ(b) is the collection of all inventory-positions attainable after the lapse of one day by an entrepreneur whose initial holdings are b; similarly, φ n (b) is the collection of all inventory-positions attainable after the lapse of n days by an entrepreneur whose initial holdings are b. Thus b c signifies that the inventory-position b is preferable to the inventory-position c in the sense that after the lapse of a certain number of days any position which could have been attained starting with c could have been bettered by a position attainable with the start b (at any rate, if c is diminished by any arbitrarily small percentage ɛ). Thus, any long-term inventory target attainable from c can be bettered if one s start is b. Hence, b c means that b is preferable to c in the long run. It is heuristically plain that the relationship b c is transitive. The following lemma states this simple fact formally.

8 8 JACOB T. SCHWARTZ Lemma If b c and c d, then b d. Proof: By hypothesis, and by Definition 4 and the remark which follows it, there exists a finite number n(ɛ) such that n n(ɛ) implies that (1 + ɛ)φ n (b) φ n (c) and (1 + ɛ)φ n (c) φ n (d). Thus (1 + ɛ) 2 φ n (b) φ n (d) for n n(ɛ). Q.E.D. Corollary. The relationship b c is reflexive, transitive, and symmetric. If b c, then b d and c d are equivalent. Lemma 5 shows that the relationship b c establishes an ordering between vectors; the central fact which we now wish to establish is that this ordering is a linear ordering, i.e., that it enables us to compare any two nonnegative vectors with the result that exactly one of b c, b c, c b holds in every case. Heuristically, this means that of two inventory positions b and c, either one is distinctly better (in the heuristic sense explained above) or both are equally good. This fact and somewhat more are stated in the following main theorem. Theorem There exists a positive vector v 0 such that the statement b c is equivalent to the statement v 0 b v 0 c. Thus, for any pair b and c of nonnegative vectors, exactly one of the statements b c, b c, c b is true. The proof of Theorem 6 will be given in what is to follow. Theorem 6 may be taken as establishing the existence of an exhaustive ranking of inventory positions as more or less favorable. Making the heuristically plausible assumption that an entrepreneur always strives to attain the most favorable position open to him, we arrive at a description of his optimal plan. Formal statements and results are contained in the following definition and in Theorem 9 below. Definition A sequence of vectors b 1, b 2, is an optimal path beginning at b if and only if b 1 = b; b n+1 φ n (b), n 1, and b n φ n (b) for arbitrarily large integers n. Our second main result will characterize the set of all optimal paths beginning at b. In order to do this, we need a number of preliminary

9 11. A MODEL OF LIQUIDITY PREFERENCE 9 definitions. The following definition extends the definition of dom(a) to arbitrary matrices. Definition 11.8a. Let A be a matrix. Then dom(a) is the maximum of the absolute value of all the (real or complex) eigenvalues of A. Definition 11.8b. Let M 1 M n+1 be the matrices of (11.5) (11.6). Then the optimal matrices are those M j for which the quantity dom(m j ) attains the value K = max 1 i n+1 dom(m i ). We may now state the second main result. Theorem The sequence of vectors b 1, b 2, is an optimal path beginning at b if and only if b 1 = b and b k+1 has the form (11.12) b k+1 = M 1 r M n+1 r n+1 where (11.13) r j 0, j = 1,, n + 1; r r n+1 = b k and where in addition we require that all components of r j but the jth and (n + 1)st vanish if M j is optimal, all components of r j but the jth vanish if M j is nonoptimal, and r n+1 = 0 if M n+1 is nonoptimal. Corollary. Any finite sequence b 1,, b m satisfying the conditions of the previous theorem for k + 1 m may be extended to an optimal sequence. If we examine the definition of the matrices M j as given by formulae (11.5) (11.6) in the light of Theorem 9, making use of the fact that b c means that b is bound to be preferable to c in the long run, we recognize the heuristic significance of Theorem 9 to be as follows: the only sound procedure for an entrepreneur whose holdings are b is to (1 ) offer the whole of each of his commodity stocks for sale; (2 ) use the whole of his cash holdings for the expansion of those stocks whose sale yields the greatest rate of profit; here, the money commodity C n+1 is to be considered as yielding the rate of profit s.

10 10 JACOB T. SCHWARTZ Theorem 9 then merely states a principle of investment practice so familiar as to be cliché. What is somewhat new is the derivation of this principle from the rather general definition of preference given by Definitions 1 and 4. We should emphasize that in (2 ) of the preceding paragraph, the rate of profit yielded by sale of a commodity C j is to be defined as the dominant of the 2 2 matrix [ ] 0 η j (11.14). σ j τ j (1 σ j ) The dominant is the unique positive root of the equation (11.15) λ 2 j λ j τ j (1 σ j ) σ j η j = 0; from this it is plain that λ increases with η and τ. Moreover, if η 1, then since η 2 ητ(1 σ) ση η(1 τ(1 σ) σ) > 0, we have λ < η: similarly λ > τ(1 σ) and hence (11.16) λ/ σ = (η λτ)/(2λ τ(1 σ)) > 0. Thus if η 1, λ increases with σ also. If σ = 0 then λ = τ; hence, as σ 0, λ τ. Consequently, Theorem 9 tells us that if (heuristically speaking) the stocks in every line of industry are sufficiently large so that each parameter σ j is distinctly reduced from 1, and if in addition the rate τ 1 j of obsolescence in each line of industry is sufficiently large so that the eigenvalues λ j of (11.15) are all less than (1 + s), the only prudent action may be to sell off stocks and hold the bond-money commodity C n+1 ; this can be the case even if the rate s of interest is zero, provided that the rates τ 1 j of obsolescence are sufficiently high. Theorem 9 thus gives a basis in efficiency economics to the decision to reduce stocks which played an important but unanalyzed role in the theory of Lectures We now intend to give the proof of Theorems 6 and 9; unfortunately, the proof involves a surprising mass of technical detail. We shall begin

11 11. A MODEL OF LIQUIDITY PREFERENCE 11 the proof in the present lecture, but shall only be able to conclude it in the following lecture. 3. Proof of Theorem 6. Initial Part In proving Theorem 6, we shall have use for some results from the theory of convex sets, which we shall now state, referring the reader to the monograph of Bonessen-Fenchel, Konvexe Körper, for proofs. The condition that a set E be convex may be written as te + (1 t)e E, 0 t 1. Let us recall that the closure of a set E of vectors is the set of all vectors b such that b is the limit of a sequence of vectors b n in E. The closure of an arbitrary set is easily seen to be a closed set. Lemma Let E and F be convex sets. Then (a) te is convex for each t 0 (b) E + F is convex (c) the closure of E is convex (d) if {E n } is a sequence of convex sets, and E n E n+1 for all n, then n E n is a convex set. Proof: For (a) and (b), see Bonessen-Fenchel, pp. 29 and 30; for (c) note that if b and c are in the closure of E, then b = lim n b n and c = lim n c n, where b n and c n belong to E. Thus, if 0 t 1, we have tb + (1 t)c = lim n (tb n + (1 t)c n ). Since E is convex, tb n + (1 t)c n lies in E, and hence tb + (1 t)c lies in the closure of E. This proves (c). To prove (d), note that if b n E n and c n E n, then, since E n E n+1 there exists an integer m such that b E m and c E m. Thus, if 0 t 1, tb + (1 t)c E m n E n. Q.E.D. sets. The following is a useful definition of convergence for closed convex

12 12 JACOB T. SCHWARTZ Definition Let K n, n 1, and K be closed convex sets. Then we write lim n K n = K or K n K, and say that the sequence K n approaches K, if for each ɛ > 0 there exists an n(ɛ) such that for all n n(ɛ) we have (11.17) K n K + S ɛ and K n + S ɛ K, S ɛ denoting the sphere of radius ɛ. Lemma Let K n, K, E n, E (n 1) be bounded closed convex sets. Then if K n K and E n E, it follows that tk n tk for all real t 0, and that K n + E n K + E. For a proof, see Bonessen-Fenchel, p. 35. Lemma Let E n, n 1, be a sequence of bounded closed convex sets such that E n E n+1. Then, if n E n is bounded, and if E denotes the closure of n E n, we have E n E. Proof: Note first that it follows from Lemma 10 that the set E is convex (and, of course, closed and bounded). Since E n E for all n, we have only to prove that for each ɛ there exists an n sufficiently large so that E E n +S ɛ. If this is false, then for each n there exists a vector p n E whose minimum distance from the set E n exceeds ɛ. Since E is bounded, we may find a subsequence p ni of p n which converges to a vector p; plainly, the minimum distance from p to E ni exceeds ɛ/2 for all sufficiently large i. Since E n+1 E n, it follows that E, which is the closure of n E n, is also the closure of i j E ni for each j no matter how large; thus the minimum distance from p to E is at least ɛ/2. But, since p is the limit of a sequence of vectors in E, this is impossible. Q.E.D. After these necessary generalities, we begin to build up the statements more directly required for the proof of Theorem 6, in a series of lemmas. Lemma The relationship defined in Definition 4 depends upon the matrices M 1,, M n+1, but remains invariant if M 1,, M n+1 are all multiplied by a common positive factor.

13 Proof: 11. A MODEL OF LIQUIDITY PREFERENCE 13 The first part of the statement is obvious from Definition 4 and the definition (11.7) of the map φ. Formula (11.7) also makes it plain that if M 1,, M n+1 are all multiplied by a common factor t, then the map φ is changed into tφ, and thus the set φ n (b) is changed into the set t n φ n (b). Definition 4. Q.E.D. The present lemma then follows at once from Lemma 14 shows that in proving Theorem 6 we may assume without loss of generality that max j dom(m j ) = 1. This will be assumed throughout the remainder of the present section. A certain subset of the matrices M j will have the distinguishing property dom(m j ) = 1; these are the matrices which are optimal in the sense of Definition 8. By renumbering the commodities C 1,, C n, we may suppose without loss of generality that set of optimal matrices is either the set M m,, M n or the set M m,, M n+1. By examining the form (11.5) of the matrices M j, remembering that the matrices M j initially introduced are first to be multiplied by a common positive factor t (1+s) 1, we see at once that if M j, j n, is an optimal matrix we have 1 = tτ j (1 σ j )+t 2 σ j η j ; if M j is not optimal, we have 1 > tτ j (1 σ j ) + t 2 σ j η j. Similarly, the matrix M n+1 is optimal if and only if (M n+1 ) n+1,n+1 = 1. The form (11.5) of the matrices M j also makes evident the following lemma. Lemma Each optimal matrix M j has a unique fixed vector u j. All components of this vector except the jth and (n + 1)st vanish. Next we use the form (11.5) of the matrices M j to prove Lemma (a) There exists a positive vector ω such that ω M j ω, j = 1,, n + 1. (b) For any vector b 0, the set φ n (b) remains bounded. Proof: Note first that (b) follows from (a). Indeed, if c φ(b), then, by the definition (11.7) of φ(b) and by (a), we have ω c ω b. By induction, this same inequality must hold for each α φ n (b), proving (b). To prove (a), we let ω n+1 = 1, and then define ω j for each j between 1 and n by requiring that (1, ω j ) be a dominant eigenvector of the transpose of the transformation [a, b] [tσ j b, tη j a + tτ j (1 σ j )b].

14 14 JACOB T. SCHWARTZ Since the constants τ k are all less than 1, it follows from the form (11.5) of the matrices M j that if [ω 1,, ω n, 1] = ω, we have ω M j ω. Q.E.D. Corollary. We have ω M j u < ω u for a nonnegative vector u unless M j is an optimal matrix and all components of u but the jth and (n + 1)st vanish. Proof: It is plain from our construction that if M j is nonoptimal ω M j < ω and that even if M j is optimal (ω M j ) k < (ω ) k unless k = j or n + 1. Q.E.D. Next we give a useful algebraic property of the mapping φ. Lemma Let E 1 and E 2 be two sets of nonnegative vectors. Then φ(e 1 ) + φ(e 2 ) = φ(e 1 + E 2 ). Proof: First suppose that r 1 φ(e 1 ), r 2 φ(e 2 ). Then, by the definition (11.7) of the map φ, we may find nonnegative vectors q 1,, q n+1 and q 1,, q n+1 such that (11.18) r 1 = M 1 q M n+1 q n+1 ; r 2 = M 1 q M n+1 q n+1, while q q n+1 E 1, q q n+1 E 2. Hence (q 1 + q 1) + + (q n+1 + q n+1) E 1 + E 2, while r 1 + r 2 = M 1 (q 1 + q 1) + + M n+1 (q n+1 + q n+1). This shows that φ(e 1 ) + φ(e 2 ) φ(e 1 + E 2 ). Conversely, let r φ(e 1 +E 2 ). Then there exist nonnegative vectors s 1,, s n+1 such that s s n+1 = e 1 + e 2, where e 1 E 1 and e 2 E 2, and such that r = M 1 s M n+1 s n+1. Define vectors q k

15 inductively by putting 11. A MODEL OF LIQUIDITY PREFERENCE 15 (11.19) (q q k ) j = min((s s k ) j, (e 1 ) j ), j = 1,, n + 1, k = 1,, n + 1. Then clearly q k 0, 1 k n + 1, and q q n+1 = e 1. It is also clear from (11.19) that, for k 2, (q k ) j = 0 unless (s s k ) j > (s s k 1 ) j, and that (s k ) j (q k ) j. Since by (11.19) this last inequality also holds for k = 1, we have (11.20) s k q k, k = 1,, n + 1. Let vectors q k be defined by q k = s k q k ; then it is clear from (11.20) that q k 0. We have (11.21) Thus and n+1 n+1 n+1 q k = s k q k = e 1 + e 2 e 1 = e 2. k=1 k=1 k=1 r 1 = M 1 q M n+1 q n+1 φ(e 1 ) r 2 = M 1 q M n+1 q n+1 φ(e 2 ). Since it is plain that r = r 1 + r 2, we have φ(e 1 + E 2 ) φ(e 1 ) + φ(e 2 ). Q.E.D. Lemma If E is convex, then φ(e) is convex. Proof: By the preceding lemma we have tφ(e) + (1 t)φ(e) = φ(te + (1 t)e) φ(e) for 0 t 1 since E is convex; thus φ(e) is convex. Q.E.D. We may now argue as follows. By Lemma 15, each optimal matrix M j has a unique fixed vector u j. The definition (11.7) of φ makes it plain that φ(u j ) M j u j = u j. Thus, inductively, φ n (u j ) φ n 1 (u j ). By the preceding lemma, all the sets φ n (u j ) are convex. By Lemmas 13 and 16(b), the sequence of sets φ n (u j ) converges to a bounded closed convex set K j ; K j is the closure of n φ n (u j ). The next lemma contributes

16 16 JACOB T. SCHWARTZ an essential bit of information by showing that all the sets K j substantially the same. are Lemma (a) If ω is the vector of Lemma 16, and the fixed vectors u j of the optimal matrices M j are normalized by the condition ω u j = 1, then all of the sets K j are identical. (b) φ(k j ) = K j. (c) The set K j contains the vector 0; thus tk j K j for 0 t 1. Proof: We first put the definition (11.7) of the mapping φ into a more convenient form. Let µ denote an arbitrary nonnegative (n+1) (n + 1) matrix whose entries µ ij satisfy the condition (11.22) n+1 µ ij = 1, i = 1,, n + 1. j=1 Let Σ j (µ), j = 1,, n + 1, be the mapping which takes a vector b into the vector Σ j (µ)b whose components are (11.23) [µ 1,j b 1,, µ n+1,j b n+1 ]. Then plainly n+1 j=1 Σ j(µ)b = b, and (11.7) shows that φ(b) may be written in the form (11.24) φ(b) = { n+1 } M j Σ j (µ)b. µ Thus, if we let j=1 n+1 (11.25) M(µ) = M j Σ j (µ), we have j=1 (11.26) φ(b) = µ M(µ)b. Since the mappings M(µ) are continuous, it follows from this last formula that M(µ)K j = M(µ) lim n φ n (u j ) lim n M(µ)φ n (u j ) lim n φ n+1 (u j ) K j ;

17 11. A MODEL OF LIQUIDITY PREFERENCE 17 thus M(µ)K j K j, and hence by (11.26) we have (11.27) φ(k j ) K j. Since u j φ(u j ) φ 2 (u j ) K j, it follows that u j K j, and thus φ(u j ) φ(k j ) and u j φ(k j ). Inductively from (11.27) we have φ n (K j ) K j ; thus φ n (u j ) φ n+1 (K j ) φ(k j ). Letting n in this last inclusion, it follows that K j φ(k j ). Combining this with (11.27), statement (b) of the present lemma follows. It is plain from the form (11.5) of the matrices M j that φ(u j ) includes a vector of the form [c 1,, c n, 0]. Thus φ n+1 (u j ) includes [τ n 1 c 1,, τ n n c n, 0]. Since τ j < 1, it follows on letting n that 0 K j, proving (c) of the present lemma. To prove (a) of the present lemma, we shall show that if M j and M k are optimal matrices and u j and u k the corresponding fixed vectors, then u j K k. Using (b) it will then follow that φ n (u j ) K k, and, letting n, that K j K k. Statement (a) will then follow at once by symmetry. If we renumber the commodities C m,, C n, we may suppose without loss of generality that k = n and j = n + 1 or j = n 1. We shall treat only the case j = n 1, leaving it to the reader to elaborate the very similar and somewhat simpler details of the case j = n + 1. Let ω be the vector of Lemma 16(a). If we examine the construction of the vector ω as described in the first paragraph of the proof of Lemma 16, and note the structure (11.5) of the matrices M j, we find that ω 0 = [0,, ω n, 1] is a fixed vector of M n, i.e., that ω 0M n = ω 0. Similarly, ω 00 = [0, 0,, ω n 1, 0, 1] is a fixed vector of M n 1. Thus, if u is a vector all of whose components except the nth and (n + 1)st vanish, we have ω M n u = ω 0M n u = ω 0 u = ω u. Similarly, if u is a vector all of whose components except the (n 1)st and (n + 1)st vanish, ω M n 1 u = ω u. Let M = M n 1 P 1 + M n P 2

18 18 JACOB T. SCHWARTZ where P 1 and P 2 are the projection operators defined by (11.28) P 1 [u 1,, u n+1 ] = [u 1,, u n 1, 0, 0] P 2 [u 1,, u n+1 ] = [0,, 0, u n, u n+1 ]. It is then clear from (11.7) that Mu φ(u) for each nonnegative vector u. Moreover, from what has been proved just above, ω Mu = ω u if all the components of u except the (n 1)st, nth, and (n + 1)st vanish. It follows from this, from (11.28) and from (11.5), and from the property of the vector u n 1 given by Lemma 15 that ω M m u n 1 = 1 for each m. According to (11.28) and the form (11.5) of the matrices M j, the (n 1)st component of M m u n 1 converges to zero, while the components 1 through n 2 are identically zero. A subsequence of this bounded sequence of vectors necessarily converges to some vector v; this vector v has all components but its nth and (n + 1)st equal to zero, while ω v = 1 and v lies in the closed convex set K n 1. Equation (11.15) for the eigenvalues of the 2 2 matrix (11.14) has clearly two distinct roots, one positive and one negative, the positive having the larger absolute value unless the square of the matrix is a constant. Since we have normalized our matrices M j in such a way that dom(m j ) = 1 if M j is an optimal matrix, it follows that (if M j is an optimal matrix) the corresponding matrix (11.14) has either the twoby-two identity matrix as its square, or has two eigenvalues of which one is +1 and the other is less than 1 in modulus. If x and y are the two eigenvectors of this 2 2 matrix, which we shall call N, x being the fixed vector, it follows in consequence that n 1 (11.29) lim n n 1 N k (αx + βy) = αx. k=0 Thus, inspecting the form (11.5) of the matrix M n, it follows that m 1 (11.30) lim n m 1 (M n ) k v exists, and is a fixed vector u of the matrix M n. Moreover, since we have as above that ω (M n ) k v = 1 for all k, it follows that ω u = 1. k=0

19 11. A MODEL OF LIQUIDITY PREFERENCE 19 Thus, since the fixed vector u n of M n is unique by Lemma 15 except for normalization, it follows that u = u n. Since we have shown above that v K n 1 and since φ(k n 1 ) = K n 1 by (b) while M n c φ(c) by (11.7), it follows from (11.30) that u n K n 1. As we have remarked above, this fact implies (a). Q.E.D. We pause at this point. The proof of Theorems 6 and 9 will be completed in the next lecture.